Cosmological Principle

The Cosmological Principle says that the universe is homogenous and isotropic. Doesn't this imply that our universe cannot be in finite size, but is finiteless? If it has a boundary, how can then the cosmological principle still be true for those heavenly bodies residing at the boundary of the universe?

If the universe has no boundary, how can we have R(t), where R is the radius of the universe in the Robertson-Walker metric? Or even in determining the future of our universe, for the different k values, 0, 1, and -1, eg, expanding forever, or Big Crunch etc.?

If the universe has no boundary, how can we have R(t), where R is the radius of the universe in the Robertson-Walker metric? Or even in determining the future of our universe, for the different k values, 0, 1, and -1, eg, expanding forever, or Big Crunch etc.?

R is not the radius of a universe ball taken from some center, it is the radius of curvature; ie., it would be the radius of the 3-sphere if the universe was a 3-sphere (spheres don't have boundaries). Similarly, expansion/contraction does not necessitate a boundary. Ie., increasing/decreasing the radius of a sphere causes the points of the sphere to move away/towards each other isotropically. The singularity referred to in the Big Bang is not the point singularity of a black hole; it is rather a singularity, "everywhere".

The Cosmological Principle says that the universe is homogenous and isotropic. Doesn't this imply that our universe cannot be in finite size, but is finiteless? If it has a boundary, how can then the cosmological principle still be true for those heavenly bodies residing at the boundary of the universe?

The Cosmological Principle says that the universe is homogenous and isotropic. Doesn't this imply that our universe cannot be in finite size, but is finiteless? If it has a boundary, how can then the cosmological principle still be true for those heavenly bodies residing at the boundary of the universe?

You are right and wrong. You are right in saying that the cosmological principle can't apply in a universe with a boundary. You are wrong in saying that a cosmological universe can't apply in a finite universe. Why? Because a finite universe does not necessarily have a boundary. A closed universe (k = 1) is one example of this. A flat universe (k = 0) with a multiply-connected topology (e.g. a flat torus, T^2) is another example.

If the universe has no boundary, how can we have R(t), where R is the radius of the universe in the Robertson-Walker metric? Or even in determining the future of our universe, for the different k values, 0, 1, and -1, eg, expanding forever, or Big Crunch etc.?

R(t) in the Robertson-Walker metric is not the radius of the universe. It is the scale factor. It is a measure of how distances scale with time in an expanding (or contracting) universe. If R(t1)/R(t0) = 2, where t1 is some time later than t0, and you have two galaxies seperated by 100 million light years at t = t0, and both have negligible peculiar motions, then these two galaxies will be seperated by 200 million light years at t = t1.